The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. The construction first establishes the circumcenter and then draws the circle. circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. This page shows how to construct (draw) the circumcircle of a triangle with compass and straightedge or ruler. This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
The image below is the final drawing above with the red labels added.
Note: This proof is almost identical to the proof in Constructing the circumcenter of a triangle.
|1||JK is the perpendicular bisector of AB.||By construction. For proof see Constructing the perpendicular bisector of a line segment|
|2||Circles exist whose center lies on the line JK and of which AB is a chord. (* see note below)||The perpendicular bisector of a chord always passes through the circle's center.|
|3||LM is the perpendicular bisector of BC.||By construction. For proof see Constructing the perpendicular bisector of a line segment|
|4||Circles exist whose center lies on the line LM and of which BC is a chord. (* see note below)||The perpendicular bisector of a chord always passes through the circle's center.|
|5||The point O is the circumcenter of the triangle ABC, the center of the only circle that passes through A,B,C.||O is the only point that lies on both JK and LM, and so satisfies both 2 and 4 above.|
|5||The circle O is the circumcircle of the triangle ABC.||The circle passes through all three vertices A, B, C|
Depending where the center point lies on the bisector, there is an infinite number of circles that can satisfy this. Two of them are shown below. Steps 2 and 4 work together to reduce the possible number to just one.